Optimization (Motivation)

Briefly, we consider the problem when a tangent line to a curve has zero slope. That is, the derivative of the curve is zero at some point.

Such a point is called a “critical point” and tells us about extreme values.

Example. Consider the function f(x)=x^{2}-2x+1. We plot this out:

Now, we can consider the tangent line at, say, x=1. We draw this tangent line in red:

Observe every point on the line (x,f(x)) has the property that f(1)\leq f(x). In other words, f(1) is a “Global Minimum” (it’s less than every other f(x) if x\not=1).

How can we detect such “extrema” (i.e., “maxima” and “minima”)? Well, the tangent line has zero slope. Equivalently: f'(x_{0})=0 when the point (x_{0}, f(x_{0})) is an extreme point.

Sometimes these “extrema” points are local. For example, plotting f(x)=x^{-1}\cos(3\pi x) when 0.1\leq x\leq1.1:

So how do we detect extrema?

Well, we take our function g(x), take its derivative, then find points x_{0} such that g'(x_{0})=0.

This is the general routine, but lets leave the motivating post with a few questions…

Problem: as we can see with the function x^{-1}\cos(3\pi x), some extrema are local while others are global. How do we
(a) Determine if an extrema is a maximum or minimum?
(b) Determine if the extrema is local or global?

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
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2 Responses to Optimization (Motivation)

  1. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  2. Pingback: [Index] Calculus in a Single Variable | My Math Blog

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